Finding the geodesic function for scalar * function= scalar

In summary, the integral in this expression represents a time-like curve with a lower left and upper right corner, and the solution for the function fgf can be found by solving a differential equation.
  • #1
metric tensor
2
0
56baaad965b3374c53c31863fac4d60e.png

In this expression the junk on the left is a scalar. The stuff before the integral is another scalar. The integral is a time-like curve between x1 and x2 and at imagine fgf(x1) is a lower left corner of the rectangle and fgf(x2) is the upper right corner and x2-x1 is the length of the base of the rectangle. I know the geodesic is the shortest distance curve paramaerized by this on the condition that the area (the integral) times the scalar1 = scalar2. The solution isn't a strait line but could be some curved function. How do I find the function fgf ?
Do i need to find the 2x2 metric tensor for the 2-d paramaterization curve and solve a pair of differential equations like here ?

http://count.ucsc.edu/~rmont/classes/121A/polarChristoffelE_Lag.pdf
 
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  • #2
Unfortunately, it is not possible to answer this question without knowing more details about the specific problem. However, in general, if you have an integral equation of the form:scalar1 = scalar2*Integral[fgf(x), x1, x2]Then you can solve for fgf by taking the integral of both sides and rearranging the equation:fgf(x) = (scalar1/scalar2)*1/(x2-x1)This should give you the function fgf that you need.
 

1. What is a geodesic function?

A geodesic function is a mathematical function that describes the shortest path between two points on a curved surface, such as a sphere or a curved space. It takes into account the curvature of the surface to determine the most efficient path.

2. How is a geodesic function different from a regular function?

A regular function operates in a flat, Euclidean space and does not take into account any curvature. A geodesic function, on the other hand, considers the curvature of the surface it is defined on and calculates the shortest path between two points accordingly.

3. What is a scalar * function = scalar equation?

A scalar * function = scalar equation is a mathematical equation that involves a scalar quantity (a single number) multiplied by a function (a mathematical rule that operates on an input to produce an output) which results in another scalar quantity as the final answer.

4. How do you find the geodesic function for a given scalar * function = scalar equation?

To find the geodesic function for a given scalar * function = scalar equation, you will need to use advanced mathematical techniques such as calculus and differential equations. These techniques can help you solve for the geodesic function that satisfies the equation and meets any additional criteria, such as being the shortest path between two points on a curved surface.

5. What are some real-world applications of finding the geodesic function for scalar * function = scalar?

The concept of geodesic functions has a wide range of applications in various fields, such as physics, engineering, and computer graphics. For example, in physics, geodesic functions can be used to describe the motion of objects in a gravitational field. In engineering, they can be used to optimize the design of structures that are subject to external forces. In computer graphics, geodesic functions can help create realistic 3D models of curved surfaces and simulate the movement of objects on those surfaces.

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